↑

Properties

Label 2-1775-1775.1561-c0-0-1
Degree $2$
Conductor $1775$
Sign $-0.829 - 0.558i$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.557i)2-s + (−0.242 + 0.747i)3-s + (−0.0314 + 0.0968i)4-s + (0.936 + 0.351i)5-s + (−0.230 − 0.708i)6-s + (−0.322 − 0.993i)8-s + (0.309 + 0.224i)9-s + (−0.913 + 0.252i)10-s + (−0.0647 − 0.0470i)12-s + (−0.490 + 0.614i)15-s + (0.718 + 0.521i)16-s − 0.362·18-s + (0.0829 + 0.255i)19-s + (−0.0634 + 0.0795i)20-s + 0.820·24-s + (0.753 + 0.657i)25-s + ⋯
L(s)  = 1  + (−0.766 + 0.557i)2-s + (−0.242 + 0.747i)3-s + (−0.0314 + 0.0968i)4-s + (0.936 + 0.351i)5-s + (−0.230 − 0.708i)6-s + (−0.322 − 0.993i)8-s + (0.309 + 0.224i)9-s + (−0.913 + 0.252i)10-s + (−0.0647 − 0.0470i)12-s + (−0.490 + 0.614i)15-s + (0.718 + 0.521i)16-s − 0.362·18-s + (0.0829 + 0.255i)19-s + (−0.0634 + 0.0795i)20-s + 0.820·24-s + (0.753 + 0.657i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $-0.829 - 0.558i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (1561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ -0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7744585930\)
\(L(\frac12)\) \(\approx\) \(0.7744585930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.936 - 0.351i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.766 - 0.557i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.242 - 0.747i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.0829 - 0.255i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.615 - 1.89i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.217 + 0.157i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 0.0897T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.0725 + 0.0527i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.340 - 1.04i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.556 + 1.71i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.635 + 0.462i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.553700613099814683975765493895, −9.223835245608920581778073123612, −8.345024222455433135819071270470, −7.34209053867595191125668210569, −6.84013630764801160921878216035, −5.82907489599340561092202668551, −5.08447744183169048820038239991, −3.99813456492270328220739261023, −3.07690694566757263618662782487, −1.62100413878946381715518386359, 0.831951050785583415141396623240, 1.80448509490531286870107069438, 2.57982265425004249575707229183, 4.17968568617973246841676421543, 5.33081953449651350211078681772, 5.97023470712367842808221688964, 6.74421082065537831778671150803, 7.74108063800905653871373548091, 8.537870701930884385960813269279, 9.392221631442647354571902220533

Graph of the $Z$-function along the critical line